MR, R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 559 
will represent in general a small spherical circle, whose coordinates are given by 
d\J/~ drjf dy/s dyjr 
d% dr] 0a> K 
abed a\ + bk 2 A ck 3 A dk^ 
. . . . (169) 
by equation (165), 
153. Given any two circles whose coordinates are (£, y, £, w), (£', y, &/), their 
power 7 r is given by 
'S, 1, 2, 3, 4\ 
U [ S', 1, 2, 3, 4/ °’ 
or 
7 r 
• K —^ 0 ^+T- 0 ^+"^; .( 17 °) 
df) 1 b 0 £ 1 0 CO 
and the radius r of the circle (£p 4 &j) will consequently be given by, 
— tan 3 r.K=2 t//(£ 77 , &>). . . . 
154. It follows that the radius of the circle 
ax- f- by-\-cz-\-dw— 0, 
(171) 
will be given by 
a l,l> 
tt l, 2> 
«L3> 
a i,n 
« 
a 2, Is 
a% 2 ? 
a 2,3> 
a 2,4J 
0 
a 3,l’ 
^3.2’ 
a 3,3> 
a 3,4> 
c 
%,i> 
2> 
a 4 3> 
a 4r» 
d 
a, 
G 
M 
= 0 ; 
(172) 
where 
M = — “(a^ + bk z + ck 3 + dk,)\ 
and where a hl , a La , &c., are the coefficients in the equation of the absolute, so that 
xjj(xyzw) ^a L pc 3 d-2« li3 a’7/+ . . . 
155. Again, the power of the circle (£y]C<o) with respect to the circle 
ax-\- by-\-oz-\- dw= 0 , 
is clearly given by 
TT = 
a% + br) + c£+dco 
ak x A bk% A dk 3 A dk^ 
■ (173) 
